Задача № 415. Найти значение следующего выражения:

\[\lim\limits_{x\to\infty}\frac{(x-1)(x-2)(x-3)(x-4)(x-5)}{(5x-1)^5}.\]

Решение.

\[\frac{(x-1)(x-2)(x-3)(x-4)(x-5)}{(5x-1)^5} = \\ = \frac{x-1}{5x-1}\cdot\frac{x-2}{5x-1}\cdot\frac{x-3}{5x-1}\cdot\frac{x-4}{5x-1}\cdot\frac{x-5}{5x-1} = \\ = \frac{1}{5^5}\cdot\frac{5(x-1)}{5x-1}\cdot\frac{5(x-2)}{5x-1}\cdot\frac{5(x-3)}{5x-1}\cdot\frac{5(x-4)}{5x-1}\cdot\frac{5(x-5)}{5x-1} = \\ = \frac{1}{5^5}\cdot\frac{5x-1-4}{5x-1}\cdot\frac{5x-1-9}{5x-1}\times \\\times\frac{5x-1-14}{5x-1}\cdot\frac{5x-1-19}{5x-1}\cdot\frac{5x-1-24}{5x-1} = \\ = \frac{1}{5^5}\cdot\left(1-\frac{4}{5x-1}\right)\cdot\left(1-\frac{9}{5x-1}\right)\times \\\times\left(1-\frac{14}{5x-1}\right)\cdot\left(1-\frac{19}{5x-1}\right)\cdot\left(1-\frac{24}{5x-1}\right).\]

Таким образом,

\[\lim\limits_{x\to\infty}\frac{(x-1)(x-2)(x-3)(x-4)(x-5)}{(5x-1)^5} = \\ = \lim\limits_{x\to 0}\left[\frac{1}{5^5}\cdot\left(1-\frac{4}{5x-1}\right)\cdot\left(1-\frac{9}{5x-1}\right)\times \\\times\left(1-\frac{14}{5x-1}\right)\cdot\left(1-\frac{19}{5x-1}\right)\cdot\left(1-\frac{24}{5x-1}\right)\right] = \\ = \frac{1}{5^5}\cdot 1 \cdot 1 \cdot 1 \cdot 1 \cdot 1 = 5^{-5}.\]

Ответ. \(5^{-5}\).

• < Назад
• Вперёд >